First, being shy and awkward as a beginning teacher is natural and (very) common. Many academics, in fact, are quite introverted. But practice will bring more comfort and more skill. Asking local colleagues about such things can also sometimes get good advice.
If you can't answer a question in class, don't try to fake it. It is ok to say "I need to think about that and will get back to you" and then bring it up again in the next lecture. As I said in a comment, however, if one student has a question then it is likely that others have the same question. I was once praised by my peers for asking so many questions in class. (My mom, however, thought I was a pest.)
Oh, and make a note so you don't forget to come back to it.
Your fallback on examples is a good start, but not enough as you have noted.
One trick is to ask a questioning student what their understanding is and see if you can find the flaw. Repeating the same explanation of a topic is likely to get the same result, so you need variations on a theme, which should come with practice. But it is easier to see where the student goes wrong and make corrections. Alternatively, ask them where they think they "lost you" in your explanation. You may need to encourage them to "work with you" toward a common and correct understanding.
Math, being axiomatic and relying on complex definitions, is especially difficult for students as the way we write and speak isn't typical language usage, with a lot of conditional phrases, each of which needs to be understood and melded into a whole. The definition of the limit, for example, is very difficult for novices. Assure them that this stuff is hard but can be learned with effort. The conciseness of mathematical explanations is also a difficulty for many students. Metaphor and analogy can help.
One bit of technology might work for you. I used to use a simple mailing list to which all students were subscribed members. Anyone on the list could ask a question there at any time (24/7) and anyone could answer the questions of others. Solutions to exercises were off limits, but questions about exercises were fine. I would answer those questions that others didn't, though it was most of them.
The advantage of the system was that it was always available but asynchronous, so I could think a bit before I answered. Also, everyone could see every question and every answer. For math, you might want something that could parse and display expressions: (Markdown or better). I found this mailing list one of the most important tools in my bag of tricks. Perhaps your IT department can set you up.
Note two things. If you are a doctoral student (or professor) then most of your students are not like you. They probably aren't driven to learn mathematics. They aren't as immersed in it as you are. The second thing is that people learn differently (and differently from yourself) so that variety of explanations is vital to develop. Some can handle a logic based response, some need metaphor, some need visual cues, some need it written out in detail. The all-hands mailing list is good for a lot of this.
Also note that it will help your students to learn if you give them lots of exercises but also extensive feedback on their "solutions". Grades aren't enough. I found this do-able with 30 students.
And, if you can arrange to get feedback on your teaching, perhaps by asking a faculty member to sit in and give you hints later, you will probably progress faster. Practice and feedback are important to many thing, including learning and teaching.
